Efficient adjustment sets for population average treatment effect estimation in non-parametric causal graphical models

The method of covariate adjustment is often used for estimation of population\naverage treatment effects in observational studies. Graphical rules for\ndetermining all valid covariate adjustment sets from an assumed causal\ngraphical model are well known. Restricting attention to causal linear models,\na recent article derived two novel graphical criteria: one to compare the\nasymptotic variance of linear regression treatment effect estimators that\ncontrol for certain distinct adjustment sets and another to identify the\noptimal adjustment set that yields the least squares treatment effect estimator\nwith the smallest asymptotic variance among consistent adjusted least squares\nestimators. In this paper we show that the same graphical criteria can be used\nin non-parametric causal graphical models when treatment effects are estimated\nby contrasts involving non-parametrically adjusted estimators of the\ninterventional means. We also provide a graphical criterion for determining the\noptimal adjustment set among the minimal adjustment sets, which is valid for\nboth linear and non-parametric estimators. We provide a new graphical criterion\nfor comparing time dependent adjustment sets, that is, sets comprised by\ncovariates that adjust for future treatments and that are themselves affected\nby earlier treatments. We show by example that uniformly optimal time dependent\nadjustment sets do not always exist. In addition, for point interventions, we\nprovide a sound and complete graphical criterion for determining when a\nnon-parametric optimally adjusted estimator of an interventional mean, or of a\ncontrast of interventional means, is as efficient as an efficient estimator of\nthe same parameter that exploits the information in the conditional\nindependencies encoded in the non-parametric causal graphical model.\n